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  "body": "\n## Definition\n\nLet $n, m \\ge 1$, let $\\Omega \\subseteq \\R^n$ be a bounded [[open set|02BH3MCBQB]], and let $\\overline{\\Omega}$ be its closure.\nLet $L \\colon \\overline{\\Omega} \\times \\R^m \\times \\R^{m \\times n} \\to \\R$, written $L(x, z, p)$, be a continuous function, called the *Lagrangian density*.\nFor a [[continuously differentiable|0VW90BBSJ4]] map $u \\in C^1(\\overline{\\Omega}; \\R^m)$ with Jacobian matrix $Du$, set\n$$J(u) = \\int_\\Omega L\\big(x, u(x), Du(x)\\big) \\, dx.$$\nThe [[functional|1EQM0DJ4NC]] $J \\colon C^1(\\overline{\\Omega}; \\R^m) \\to \\R$ is the *integral functional* with Lagrangian density $L$.\nThe integrand is continuous and bounded on $\\Omega$, so the [[Lebesgue integral|DWJF7ABPR1]] is finite.\n\n## Remarks\n\nIn one variable, with $n = m = 1$ and $\\Omega = (a, b)$, the functional reads $J(y) = \\int_a^b L(x, y(x), y'(x)) \\, dx$.\nExample: $L(x, z, p) = \\frac{1}{2} \\abs{p}^2$ with $m = 1$ gives the Dirichlet energy $J(u) = \\frac{1}{2} \\int_\\Omega \\abs{\\nabla u}^2 \\, dx$.\nIt is an integral functional because this $L$ is continuous on all of $\\overline{\\Omega} \\times \\R \\times \\R^n$.\nNon-example: on $\\Omega = (0, 1)$, the expression $L(x, z, p) = 1/x$ is not defined at $x = 0$ and has no continuous extension to $[0, 1] \\times \\R \\times \\R$, and indeed $\\int_0^1 dx / x = \\infty$ for every $y$.\nThe same name is used for Lagrangians with higher derivatives, as in the [[Euler–Poisson equation|XBNAJ1RRET]], and for $u$ in [[Sobolev spaces|M4CEZ5K9HP]] under growth conditions on $L$.\n",
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